1 Introduction

To achieve greater performance of aeroengine, the overal fuel/air ratio and exit temperature of aeroengine combustor are increased gradually, resulting in the air quantity participating in combustion increasing. Therefore a great challenge is posed to the traditional combustion mode, and in order to adapt to the combustor development of high temperature rise in the future, new combustion modes are needed, such as Rich-Quench-Lean (RQL), Twin Annular Premixing Swirler (TAPS), Lean Direct Injection (LDI) and Trapped Vortex Combustor (TVC)^[1]. It is quite important to combustor design to improve the knowledge and understanding of the flow and combustion in combustor. Nevertheless it is difficult to carry out lots of experiments on flow and combustion in combustor due to the cost and cycle. CFD provides an alternative method to study the flow and combustion in combustor in recent years^[2].

Flamelet method is a simplified method that can take into account the detailed chemical mechanisms^[3], and has been developed rapidly in combustion simulation. Steady flamelet method is a widely used method at present owing to its high computational efficiency and stability. In steady flamelet method the thermochemical parameters in combustion are mapped onto several control parameters, establishing a functional relationship known as laminar flamelet library between thermochemical parameters and these control parameters, and the library is obtained by solving a set of laminar flamelet equations. The method constructing the laminar flamelet library is the major differences among different steady flamelet models, mainly including construction methods based on laminar diffusion and premixed flamelet. Verhoeven L M et al^[4] reported the comparisons between premixed and diffusion Flamelet Generated Manifolds (FGM) on the simulation of non-premixed laminar co-flow flames, concluding that the diffusion FGM has higher accuracy than premixed FGM on temperature and species mass fraction. Moin P et al^[5] adopted Flamelet/ Progress Variable (FPV) combined with Large Eddy Simulation and a detailed chemical mechanism of Jet-A to model the combustion in a Pratt & Whitney RQL combustor sector, and the results verified the applicability of FPV model in aeroengine combustor. FGM with five-dimensional lookup table developed by Donini A et al^[6] is adopted to simulate the turbulent combustion in a gas turbine model combustor, and comparison with experimental data showed that FGM had high accuracy in the prediction of velocity and temperature, but had a certain error in the prediction of CO mass fraction.

In addition to the differences in construction of laminar flamelet library, Probability Density Function (PDF) describing the interaction between turbulence and combustion also has an important effect on the prediction accuracy of flamelet models. Reference[7] adopted Steady Laminar Flamelet Model (SLFM) with the presumed PDF and transported PDF respectively to model the CH₄/Air turbulent Jet diffusion flame, and concluded that transported PDF has no obvious advantage to presumed PDF. Gupta A et al^[8] studied the combustion process of a Rolls Royce RQL combustor with FGM combined with presumed PDF and transported PDF respectively, and reached a contrary conclusion that transported PDF had a higher accuracy in the prediction of temperature, NO_x and CO. Neverthless the enormous amount of computation limits the application of transported PDF in complex geometry. In the presumed PDF, each control parameters has a different PDF type. For example the β distribution is usually as the PDF of mixture fraction, but the PDF of reaction progress variable has a variety of PDF options such as δ distribution, β distribution and Statistically Most Likely Distribution (SMLD). Furthermore the PDF type of reaction progress variable has a quite important impact on the prediction of some phenomena, such as local extinction and reignition^[9]. Ihme M et al ^{[10, 11]} studied the local extinction and reignition in Sandia Flame D and E with FPV model combined with the three kinds of reaction progress variable PDF of δ and β distribution, and SMLD respectively, and concluded that SMLD had the highest precision on the simulation of flame with strong local extinction and reignition and the δ distribution had the lowest precision. Ravikanti M ^[12] compared the accuracy of FPV model with δ and β distribution and SLFM in the liftoff height of Cabra lifted flame respectively, indicating β distribution has the highest accuracy, and moreover SLFM couldn not model the lift of flame.

In current paper, FPV model based on diffusion flamelet and FGM model based on premixed flamelet are used to model the turbulent combustion in a dual-swirler aeroengine model combustor, and in FPV and FGM models the PDF type of reaction progress variable includes δ and β distribution respectively. So the effects of the construction method of laminar flamelet library and PDF type of reaction progress variable on the combustion field of realistic aeroengine combustor are studied. A comparison between the numerical and experimental results from Coherent Anti-Stokes Raman Scattering (CARS) and Tunable Diode Laser Absorption Spectroscopy (TDLAS)^[13] verifies the accuracy of above combustion models.

2 Mathematical model 2.1 Combustion model

FPV and FGM models have the same form of laminar flamelet library in which the thermochemical parameters are the function of mixture fraction Z and reaction progress variable C. The reaction progress variable is defined as

$ C = {Y_{{\rm{C}}{{\rm{O}}_2}}} + {Y_{{\rm{CO}}}} + {Y_{{{\rm{H}}_2}{\rm{O}}}} + {Y_{{{\rm{H}}_2}}} $

(1)

However, there is a great difference in the construction of laminar flamelet library. In FPV model, the construction of laminar flamelet library is accomplished by solving a set of laminar opposed-flow diffusion flamelet equations, and in order to uniquely identify every flamelet represented by reaction progress variable, the reaction progress variable at stoichiometric condition is normalized as the flamelet parameter^[14] λ defined as

$ \lambda = \frac{{{C_{{\rm{st}}}} - {C_{{\rm{st}}, {\rm{min}}}}}}{{{C_{{\rm{st}}, {\rm{max}}}} - {C_{{\rm{st}}, {\rm{min}}}}}} $

(2)

Where, subscript st represents the stoichiometric condition.

In FMG model, the laminar flamelet library is constructed by solving a set of laminar stretchless premixed flamelet equations^[15]. In order to obtain the uniquely certain corresponding relation between reaction progress variable and other thermochemical parameters at every mixture fraction, the reaction progress variable at every mixture fraction is normalized as flamelet parameter y defined as

$ y = \frac{{C - {C_{{\rm{min}}}}}}{{{C_{{\rm{max}}}} - {C_{{\rm{min}}}}}} $

(3)

ξ is used as the general name for λ and y, and the final laminar flamelet library of FPV and FGM models is expressed as the following form

$ \phi = \phi \left( {Z, \xi } \right) $

(4)

The normalization of reaction progress variable is also convenient for the β integration of reaction progress variable. The β distribution is used as the PDF of mixture fraction, and reaction progress variable adopts the δ and β distribution as its PDF. Assuming that mixture fraction and reaction progress variable are statistically independent, the joint PDF of mixture fraction and flamelet parameter of which PDF type adopts δ and β distribution can be expressed as, respectively

$ \tilde P\left( {Z, \xi } \right) = \beta \left( {Z;\tilde Z, \widetilde {{{Z''}^2}}} \right)\delta \left( {\xi - \tilde \xi } \right) $

(5)

$ \tilde P\left( {Z, \xi } \right) = \beta \left( {Z;\tilde Z, \widetilde {{{Z''}^2}}} \right)\beta \left( {\xi ;\tilde \xi , \widetilde {{{\xi ''}^2}}} \right) $

(6)

The equation (5) and (6) are respectively used to integrate equation (4) to form flamelet lookup table employed to the simulation of turbulent combustion as shown as follow

$ \tilde \phi = \tilde \phi \left( {\tilde Z, \widetilde {{{Z''}^2}}, \tilde \xi } \right) $

(7)

$ \tilde \phi = \tilde \phi \left( {\tilde Z, \widetilde {{{Z''}^2}}, \tilde \xi {\rm{, }}\widetilde {{{\xi ''}^2}}} \right) $

(8)

Where, the processing of β distribution is accomplished by Piecewise Integration Method (PIM)^[16].

Because there are no exact equations for both λ and y, so the thermochemical parameters are remapped to the mean reaction progress variable and its variance^[12]. The turbulence is captured by Standard k-ε turbulent model, and then the corresponding transport equations for the mean mixture fraction $ {\tilde Z}$ and its variance $ {\widetilde {{{Z''}^2}}}$, mean reaction progress variable $ {\tilde C}$ and its variance $ {\widetilde {{{C''}^2}}}$ are shown in equation (9)~(12), respectively

$ \frac{{\partial \left( {\bar \rho {{\tilde u}_i}\tilde Z} \right)}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_i}}}\left[ {\left( {\bar \rho {D_Z} + \frac{{{\mu _{\rm{t}}}}}{{{\sigma _Z}}}} \right)\frac{{\partial \tilde Z}}{{\partial {x_i}}}} \right] + {S_Z} $

(9)

$ \begin{array}{*{20}{l}} {\frac{{\partial \left( {\bar \rho {{\tilde u}_i}\widetilde {{{Z''}^2}}} \right)}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_i}}}\left[ {\left( {\bar \rho {D_Z} + \frac{{{\mu _{\rm{t}}}}}{{{\sigma _Z}}}} \right)\frac{{\partial \widetilde {{{Z''}^2}}}}{{\partial {x_i}}}} \right] + }\\ {2\frac{{{\mu _{\rm{t}}}}}{{{\sigma _Z}}}{{\left( {\frac{{\partial \widetilde {{{Z''}^2}}}}{{\partial {x_i}}}} \right)}^2} - 2\frac{{\tilde \varepsilon }}{{\tilde k}}\bar \rho \widetilde {{{Z''}^2}} + {S_{{{Z''}^2}}}} \end{array} $

(10)

$ \frac{\partial \left( \bar{\rho }{{{\tilde{u}}}_{i}}\tilde{C} \right)}{\partial {{x}_{i}}}=\frac{\partial }{\partial {{x}_{i}}}\left[ \left( \bar{\rho }{{D}_{C}}+\frac{{{\mu }_{t}}}{{{\sigma }_{C}}} \right)\frac{\partial \tilde{C}}{\partial {{x}_{i}}} \right]+\bar{\rho }{{\tilde{\dot{\omega }}}_{C}} $

(11)

$ \begin{array}{*{35}{l}} \frac{\partial \left( \bar{\rho }{{{\tilde{u}}}_{i}}\widetilde{{{{{C}''}}^{2}}} \right)}{\partial {{x}_{i}}}=\frac{\partial }{\partial {{x}_{i}}}\left[ \left( \bar{\rho }{{D}_{C}}+\frac{{{\mu }_{\text{t}}}}{{{\sigma }_{C}}} \right)\frac{\partial \widetilde{{{{{C}''}}^{2}}}}{\partial {{x}_{i}}} \right]+ \\ 2\frac{{{\mu }_{t}}}{{{\sigma }_{C}}}{{\left( \frac{\partial \tilde{C}}{\partial {{x}_{i}}} \right)}^{2}}+2\bar{\rho }\left( \widetilde{C{{{\dot{\omega }}}_{C}}}-\tilde{C}{{{\tilde{\dot{\omega }}}}_{C}} \right)-2\frac{{\tilde{\varepsilon }}}{{\tilde{k}}}\bar{\rho } \widetilde{{{{{C}''}}^{2}}} \\ \end{array} $

(12)

Where, σ_Z and σ_C have a value of 0.7 and 0.4 respectively. Due to the evaporation of Jet-A, mean mixture fraction and its variance have a mass source of S_Z and S_Z"², and mean reaction progress variable and its variance have no mass source as the definition of reaction progress variable is independent of Jet-A. In addition, due to the energy exchange between gas and liquid phase during evaporation, the enthalpy is nonconservative, and the transport equation of mean enthalpy $ {\tilde h}$ is shown in equation (13)

$ \frac{{\partial \left( {\bar \rho {{\tilde u}_i}\mathit{\tilde h}} \right)}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_i}}}\left[ {\left( {\bar \rho {D_h} + \frac{{{\mu _{\rm{t}}}}}{{{\sigma _h}}}} \right)\frac{{\partial \mathit{\tilde h}}}{{\partial {x_i}}}} \right] + {S_h} - {Q_{{\rm{rad}}}} $

(13)

Where σ_h has a value of 0.7; S_h is the energy source caused by evaporation, and Q_rad is the radiation source simulated by Optically Thin Model (OTM)^[17].

2.2 Atomization model

The exit condition of the fuel injector plays an important role in the simulation of atomization and evaporation of liquid Jet-A. The liquid Jet-A is injected into combustor through a simplex pressure-swirl injector and the atomization is modeled by tracking spray particles in a Lagrangian framework.

The breakup of liquid film formed by pressure-swirl atomizer is modeled by Linearized Instability Sheet Atomization (LISA) model developed by Schmidt D P et al^{[18, 19]}. The breakup distance of liquid film can be predicted by taking into account the formation, development and breakup of liquid film.

The further breakup of liquid droplets from breakup of liquid film is modeled by KHRT model developed by Reitz R D^[20], that simultaneously considers the Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) instability, and the breakup of droplets is the result of the competition between KH and RT instability wave.

The evaporation of Jet-A is modeled by non-equilibrium Langmuir-Knudsen model^{[21, 22]}. In this model, the temperature difference is ignored in droplet and the energy exchange between droplets and surrounding gas includes heat convection only. In addition, the influence of the non-equilibrium effect on evaporation of droplets is considered. The source of evaporation is simulated with PSI-Cell method, and then the diffusion flame surrounding droplet is ignored. To avoid the source resulted from evaporation causing numerical instability in solving the transport equations, the source is handled by under-relaxation to improve the numerical stability, and the relaxation factor has a value of 0.5.

3 Experimental and numerical setup 3.1 Experimental setup

The experimental geometry is a dual-swirler with discrete jets and radial swirler model combustor, and the liner is as shown in Fig. 1. The original combustor sector is simplified to a rectangle structure as shown in Fig. 1, and the upper and lower wall of liner are symmetry. In each wall there are a complete and two half primary holes, two complete and two half dilute holes and six rows of film cooling. The swirler cup consists of a primary swirler with eight discrete jets followed by a venturi, and a secondary swirler with ten curved blades followed by a flare, as shown in Fig. 2.

The optical experiments are carried out in this combustor in early work^{[13, 23]}, which not only provides an understanding of the flow and combustion in the aeroengine combustor but also a validation data for numerical code. In current paper, the data of temperature resulted from CARS and TDLAS is used to verify the accuracy of turbulent combustion models, and the experimental condition is showed in Table 1. The measurement points of CARS are showed in Fig. 3, including 12 locations, among which the line-averaged temperature perpendicular to the planar of Fig. 3 in P₃ and P₁₀ two locations is measured by TDLAS.

3.2 Numerical setup

The computational domain is showed in Fig. 4. Except the liner as shown in Fig. 1, there are also a diffuser, fuel injector, igniter and casing as the same as experimental setup. The coordinate origin is defined in the center of dome exit, and the coordinate direction is shown as Fig. 4. Tetrahedral mesh is used to divide this computational domain, and the total number of mesh is about 3.5 millions.

N-S equations are solved by SIMPLE algorithm in collocated mesh, and the solver has second order accuracy in space. All transport variables have Dirichlet inlet conditions except pressure which has a Neumann inlet condition, and moreover all transport variables have Neumann outlet condition except pressure which has a Dirichlet outlet condition. The two scale wall function is used as the wall function.

The Jet-A is modeled by n-C₁₀H₂₂, the chemical mechanism of which is a mechanism of 89 species and 680 reactions reduced by Luche. The exit of injector is defined at the center of exit of primary swirler, and the angle of spray cone has a value of 90° and the dispersion angle has a value of 10°.

The combustion in the aeroengine model combustor is simulated by FPV and FGM of which reaction progress variable has δ and β distributions, expressed as FPV-δ, FPV-β, FGM-δ and FGM-β, respectively.

4 Results and discussion

Table 2 shows the comparison of temperature among the above four combustion models and CARS results and the error in the 12 points. As shown in the table, FPV-β and FGM-β models have higher accuracy than FPV-δ and FGM-δ compared with CARS, indicating that reaction progress variable with β distribution taking second moment information into account can better reflect local extinction and reignition due to the interaction between swirl flow and primary jets, and hence the prediction on temperature has higher accuracy. In addition, the comparison between FPV-β and FGM-β models can be seen, FPV-β shows higher accuracy in P₁ and P₆, especially in P₁, but the results of FGM-β coincide with the results of CARS better in the rest points. P₁ is located between swirl flow and the upper primary jet where the interaction between swirl flow and primary jet is quite strong, and P₆ is located close to the reversed flow formed by the wake of primary jets getting into recirculation zone. Compared with the other locations, the flow and turbulence in P₁ and P₆ have a stronger effect on flame structure, and hence the characteristic of diffusion flame is more significant. As the lookup table of FPV-β model is constructed based on diffusion flame and contains the effect of flow and strain on structure of flame, FPV-β model can better describe the flame with large strain and has higher accuracy in the two points. In addition, there is a great of gradient of mixture fraction in the two points, and the diffusion of mixture fraction is becoming important, but the diffusion of mixture fraction is not considered in the construction of flamelet lookup table of FGM-β model. Hence FGM-β model has greater deviation than FPV-β model in the two points. P₃ is located between the upper primary jet and recirculation zone, but contrast with the other two points it is closer to the recirculation zone and the characteristic of premixed flame also plays an important effect on flame, and hence the two models have similar accuracy. P₇ is located around the reversed flow entering recirculation zone from primary jets wake, but the reversed flow has a weaker effect on P₇ compared with P₆, and hence the two models have similar accuracy. Except P₁₁ and P₁₂, the rest points are located in or closer to recirculation zone (P₂, P₅, P₉ and P₁₀), or close to the center of primary jets (P₄ and P₈), and the characteristic of premixed flame has a stronger effect, and hence FGM-β has higher accuracy. As P₁₁ and P₁₂ are located close to the wake of primary jets entering secondary zone, the flow is quite complex and hence the current turbulent model cannot predict the structure of flow accurately, resulting in large deviations of the temperature prediction also.

Table 3 demonstrates the comparison of temperature among above four combustion models and TDLAS results and the error in P₃ and P₁₀. As shown in Table 3, the results from the four combustion models are in good agreement with that of TDLAS. Compared with δ distribution, β distribution does not show higher accuracy, and only the precision of FGM-β model is slightly improved in P₃.

The temperature from TDLAS is a line integrated temperature and the line could cross multiple reaction zones that could have different combustion modes. As seen from Table 3, as the characteristic of premixed flame on the line crossing P₃ has a more important impact, hence FGM model gets higher accuracy. As the line crossing P₁₀ is closer to reversed flow entering recirculation zone of primary jets, and the characteristic of diffusion flame is more important, and hence FPV model gets higher accuracy.

Fig. 5(a) and (b) are the diameter distribution of spray resulted from FGM-β model combined with LISA and KHRT models in combustor and close to the venturi respectively, and the rest three models have similar results. The particles of large diameter displayed by red actually represent the thickness of liquid film injected from fuel injector. In current paper, in order to facilitate the writing and description of the program about spray, the liquid film is still represented by discrete droplet, and the thickness is represented by particle diameter also. The thickness of liquid fuel is quite large when it just leaves the injector, and then the thickness rapidly decrease under the effect of centrifugal force by injector and swirl flow by primary swirler together. When the breakup condition is satisfied, the liquid film is broken into discrete droplets. Because these small pieces of liquid film produced during the breakup of liquid film would polymerize into droplets with a certain diameter, and the diameter of droplets are quite larger than thickness of liquid film before breakup, the diameter of droplets rapidly increases in the breakup position, as shown as the rapid change of diameter in the zone marked by black ellipse, and the position of film breakup is 4.6 mm away from the injector exit.

The velocity and direction of droplets keep the same as that of liquid film before breakup. The droplets move along the wall of venturi and break into droplets of small diameter after collision with venturi, and the breakup is accelerated, as shown in Fig. 5(b) that after droplets collide with venturi the diameter of droplets decreases significantly from about 40μm to about 25μm. In the downstream of venturi exit, the diameter of droplets decreases rapidly from 20μm to 10μm under the effect of shear force formed by the counter-rotating primary and secondary swirl flow, and the process is accompanied by violent evaporation of droplets. Therefore the evaporation of fuel is basically completed at the downstream close to dome exit, as shown in Fig. 5(a).

Fig. 6 shows the distribution of mean mixture fraction and streamline on the plane of z=0. As shown from the distribution of streamline, the structure of flow field is asymmetric because of the fuel injector and igniter, and hence the penetration depth of the lower primary jet is larger than the upper primary jet. The angle of the upper primary jet is reduced and hence has a more obvious effect on secondary zone, resulting in the wake of primary jets deflecting the lower wall of liner. The upper wall of liner has a stronger effect on the primary zone. The four models have a little effect on the structure of flow field, and there are some differences in the wake angle of primary jets only. The wake resulted from FPV-β and FGM-β models approaches the lower wall more closely, especially for FGM-β model. As seen from the distribution of mean mixture fraction, the evaporation of fuel starts around the impingement point between spray and venturi wall, which is also the position of acceleration of droplets breakup. Nevertheless large amount of fuel evaporation occurs at the downstream of venturi exit, where the diameter of droplets becomes quite small under the effect of the counter-rotating swirl flow. The mixture fraction reaches maximum around the exit of flare, and then the mixture fraction decreases gradually due to the dilution of fresh air and burned gas. However, the zone of high mixture fraction is concentrated in the swirl flow and the shear layer between swirl flow and recirculation zone, where the mixture fraction is larger than that in recirculation zone. Because the evaporation of fuel needs the participation of surrounding air, the mixture fraction is far below 1, even in the zone of the highest mixture fraction. The mixture fraction in the most of primary zone is dramatically greater than stoichiometric mixture fraction 0.063 based on n-C₁₀H₂₂, and high mixture fraction in the primary zone is diluted by primary jets, and gradually decreases to about stoichiometric mixture fraction in secondary zone. In the exit of secondary zone the mixture fraction decreases further in the exit of secondary zone due to the dilution jets.

There is a significant difference between FPV and FGM models in the prediction of mixture fraction. In FPV model, high mixture fraction is concentrated on the zone between exit of flare and dome, but in FGM model high mixture fraction is extended upward to the venturi exit and downstream to the dome exit and moreover in the shear layer the mixture fraction resulted from FGM model is larger than that from FPV model. This indicates that the diffusion of mixture fraction modeled by FPV model is faster than FGM model. In the construction of laminar flamelet library, the diffusion of mixture fraction is ignored in FGM model and considered in FPV model, and therefore the mixture fraction in FGM model is more concentrated than that in FPV model.

As seen from the comparison between FPV-δ and FPV-β models and between FGM-δ and FGM-β models, the effects of PDF type of reaction progress variable on the distribution of mixture fraction are almost the same for FPV and FGM models. There is a great difference between splash and swirl flow at the downstream of flare for the δ and β distribution. The mixture fraction from δ distribution is obviously higher than that from β distribution. Meanwhile in the rest of primary zone the mixture fraction from β distribution is slightly greater than that from δ distribution. In the secondary zone, the mixture fraction from β distribution is more homogeneous than that from δ distribution.

Fig. 7 and Fig. 8 are the distribution of temperature and fuel mass fraction in gas phase respectively on the plane of z=0. Due to the second moment information of reaction progress variable ignored, δ distribution has a great deviation on the simulation of the unburned mixture ignited by the recirculated burned gas. Because the unburned mixture is ignited faster by the recirculated burned gas from δ distribution than that from β distribution, so there is an obvious high temperature zone in the downstream close to the venturi exit, and the high temperature zone is closer to the injector and wider than that from β distribution as shown in Fig. 7. This indicates that the edge of unburned mixture is ignited earlier in δ distribution, preventing the diffusion of the unburned mixture closer to the wall of venturi and flare to the recirculation zone, and hence the mixture fraction and fuel mass fraction from δ distribution are greater than those from β distribution. As shown in Fig. 6 and Fig. 8 respectively, the distribution of mixture fraction and fuel mass fraction from β distribution is more homogeneous than that from δ distribution around the venturi exit. Because the diffusion of unburned mixture into recirculation zone is prevented by the high temperature zone arising earlier in the simulation of δ distribution, so the unburned mixture keeps sufficient momentum and the length and scope of high mixture fraction or fuel mass fraction and low temperature zone are larger than those in β distribution. Therefore there is higher fuel/air mixture between the flash and swirl flow in the simulation of δ distribution as demonstrated by higher mixture fraction in Fig. 6, and the higher mixture fraction also leads to the temperature in this zone becoming higher in the simulation of δ distribution.

As demonstrated from the comparison between FPV-β and FGM-β models, the extended distance of the low temperature zone along flare in FPV-β model is larger than that in FGM-β model. Therefore there is larger mixture fraction and temperature between the flash and swirl flow.

As seen from the distribution of fuel mass fraction, there is a great difference between FPV and FGM models. In FPV model, the fuel is almost completely decomposed around the dome exit, but in FGM model, the decomposition distance of fuel is larger than that in FPV model, and the fuel keeps a high concentration in the downstream boundary of primary zone. As the same as the distribution of mixture fraction, FGM model ignores the diffusion between variety mixture fraction during the construction of laminar flamelet library, resulting in the fuel being more concentrated than that in FPV model. Therefore the zone containing fuel in FGM model is larger than that in FPV model.

The decomposition distance of fuel in β distribution is larger than that in δ distribution, and the fuel is extended to the shear layer between primary jets and recirculation zone along swirl flow, and the distribution of fuel in FGM-β is extended to the impingement point. There is strong exchange of mass, momentum and energy in the shear layer between swirl flow or primary jet and recirculation zone, and hence strong unsteady phenomenon exits in the shear layer, which weakens the decomposition of fuel. As β distribution can capture this unsteady phenomenon, the scope of fuel is larger, avoiding the enrichment and reaction of fuel that leads to large amount of species of large time scale, such as CO. Therefore in the primary zone and secondary zone, β distribution can obtain more homogeneous temperature distribution.

5 Conclusions

The main conclusions of this study are as following:

(1) The FPV model of which laminar flamelet library is based on the laminar diffusion flame has higher accuracy in these locations close to swirl flow and primary jet, and moreover the FGM model of which laminar flamelet library is based on the laminar premixed flame has higher accuracy in these location close to and in recirculation zone.

(2) The models in which the PDF of reaction progress variable adopts the β distribution taking into account the second moment information can effectively improve the accuracy of temperature, and the error of temperature from FPV-β and FGM-β models is almost below 5% compared with CARS in the primary zone.

(3) LISA model can predict the breakup distance of liquid film form by pressure-swirl atomizer, and the breakup distance is 4.6mm. The droplets diameter decreases rapidly to about 10μm in the downstream of venturi exit in the prediction of KHRT model.

(4) FPV-β and FGM-β taking the β distribution as the PDF of reaction progress variable can better describe the process that the unburned mixture is ignited by the recirculated burned gas, and the distribution of corresponding thermochemical parameters. The PDF type of reaction progress variable has a more important effect than the construction of laminar flamelet library in the prediction of ignition process of unburned mixture.